Coexisting Attractors, Energy Analysis and Boundary of Lü System

In this study, first, the phenomenon of multistability in the Lü system is found, which shows the coexistence of two different point attractors and one chaotic attractor. These coexisting attractors are dependent on initial conditions of the system while the parameters of the system are fixed. Then, the Lü system is transformed to a Kolmogorov-type system, which includes the conservative torque consisting of the inertial torque and the internal torque, the dissipative torque, and the external torque. Moreover, by analyzing the combination of different types of torques and investigating the cycling of energy based on the Casimir function and Hamiltonian function, the interaction between the external torque and other torques is found to be the main reason for the Lü system to generate chaos. Finally, by investigating the Casimir function, it is found that the boundary of the Lü system is only related to system parameters.

Dynamical Mechanism and Energy Conversion of the Couette–Taylor Flow

In this paper, we study the dynamical mechanism and energy conversion of the Couette–Taylor flow. The Couette–Taylor flow chaotic system is transformed into the Kolmogorov type system, which is decomposed into four types of torques. Combining different torques, the key factors of chaos generation and the physical interpretation of the Couette–Taylor flow are studied. We further investigate the conversion among Hamiltonian, kinetic and potential energies, as well as the correlation between the energies and the Reynolds number. It is concluded that the combination of the four torques is necessary to produce chaos, and the system can produce chaos only when the dissipative torques match the driving (external) torques. Any combination of three types of torques cannot produce chaos. Moreover, we introduce the Casimir function to analyze the system dynamics, and choose its derivation to formulate the energy conversion. The bound of chaotic attractor is obtained by the Casimir function and Lagrange multiplier. It is found that the Casimir function reflects the energy conversion and the distance between the orbit and the equilibria.

Mechanism and Energy Cycling of the Qi Four-Wing Chaotic System

The Qi four-wing chaotic system is transformed into a Kolmogorov-type system, thereby building a bridge between a numerical chaotic system and a physical chaotic system that is convenient for analysis when finding common ground between the two. The vector field is decomposed into four types of torques: inertial, internal, dissipative and external. The angular momentum representing the physical analogue of the state variables of the chaotic system is identified. The cycling of energy among potential energy, kinetic energy, dissipation, and external energy is analyzed. The Casimir function is employed to identify the key factors producing chaos and other dynamical modes. The system is non-Rayleigh dissipative, which determines the extremal points of Casimir function to form a hyperboloid instead of ellipsoid.

A Three-Dimensional Wave-Activity Relation for Pseudomomentum

A three-dimensional, nonhydrostatic local wave-activity relation for pseudomomentum is derived from the nonhydrostatic primitive equations in Cartesian coordinates by using an extension of the momentum–Casimir method. The stationary and zonally symmetric basic states are chosen and a Casimir function, which is the single-valued function of potential vorticity and potential temperature, is introduced in the derivation. The wave-activity density and wave-activity flux of the local wave-activity relation for pseudomomentum are expressed entirely in terms of Eulerian quantities so that they are easily calculated with atmospheric data and do not require the knowledge of particle placements. Constructed in the ageostrophic and nonhydrostatic dynamical framework, the local wave-activity relation for pseudomomentum is applicable to diagnosing the evolution and propagation of mesoscale weather systems.

TRI–HAMILTONIAN VECTOR FIELDS, SPECTRAL CURVES AND SEPARATION COORDINATES

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1-λP0) and (P2-μP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.